This book is a translation into English of Hilbert's "Theorie der algebraischen Zahlkrper" best known as the "Zahlbericht", first published in 1897, in which he provided an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht provided also a firm foundation for further research in the subject. It is based on the work of the great number theorists of the nineteenth century. The Zahlbericht can be seen as the starting point of all twentieth century investigations in algebraic number theory, reciprocity laws and class field theory. For this English edition an Introduction has been added by F. Lemmermeyer and N. Schappacher.